308 Mr Pocklington, Standing waves parallel to a plane beach 



ION, ivi.A., bt Jolin s College. 



[Received 18 January; Read 7 February, 1921.] 



Darailer^n f ^i''^ ""^ *r ' ^^^'1'' *^ investigate the standing waves 

 parallel to the shore hne m the case of an infinite fluid bounded 

 below by a plane sloping at an angle a to the horizontal We 

 rest,,ct ourselves to the case where a is a sub-multiple of a righ? 

 TrfTfi'"'" *^!.^^*^d ^f i^^ges. The conditions to be satisfied 

 are stated m section 2 and the images formed by the boundaries 

 (beach and free surface) are found in section 3. VLtiorfwe 

 c^dlw t '.t 7^r*{ /t^tial and show that it satisfierall tl 

 To^la^t,^^^^^' -^^ '''*^"^^ ^' ^^^ *^^* *^^ amplitude 

 pressSn of ?).f i "^"""V^ '^'^'^'"^ ^^^ consequence of the com- 

 vl^M-l ^' ^^ ^'* '''*^ '^^^^^^ ^^*^^) ^^ t^e ratio 



2. Let the hquid be bounded below by a plane beach sloping 

 at an ang e a ^^/2n. We take the shore fine as axis of .and us? 

 cylindrical coordmates, so that ^ = is the equation of the free 

 surface of the hqmd, and ^ ^ a is that of the beach. Let the period 

 of the standing waves be 2n/p. Then the velocity potential is of 

 are ^^^^2!?/^ ^^1+ ^^- ^^' conditions to be satisfied by O 



thei^vit:^!^ 



n■^ ^' ^^ ^l ^r.^ ^ ^i""^ ""^ ^' ^^y ^1' *hat satisfies (i) but not 

 ) we can find O, so that O, satisfies (i) and 0), + O, satisfies li 

 (It will of course satisfy (1)), and may call O, the ima|e of 0, w th 

 respect to lower boundary. Similarly, if (D, satisfies (i) but n JtTiin 

 and we find O, to satisfy (i) and such that <D, + oVsatisfies 

 (It will of course satisfy (i)) then O, may be cal ed the image ^^^^^^^ 

 with respect to the upper boundary. ^ 1 



If 



Oi = exp {- Xr sin (d ~ ^)} cos [r] + Xr cos {9 - ^)} =f{~l3 rj d) 



O2 = exp {- Xr sin (2a - /3 - ^)} cos {77 + Ar cos {2a - fi - d)} 

 = fi^^-P,V,~e)=f(P',r]\~e) say. 



